Family 1-9-7-a

A smooth complete intersection of two cubics in $\mathbb P^6$.

ℙ⁶P6 · 600.CC (preferred model)

Defining data

Ambient: ℙ⁶P6
Bundle: $(E^{3 ω_{1}})^{\oplus 2}$ 2 copies of E^(3w1)
Bundle: $O (3)^{\oplus 2}$ 2 copies of O(3)
ZL62 label: 600.CC

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7) \\ \det\mathcal{E} &\cong \mathcal{O}(6) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	7
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	9
$\chi(X, \mathrm{T}_X)$	=	-116

$\omega_X^\vee \cong \mathcal{O}_X(1)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	9
$\mathrm{c}_1^2\mathrm{c}_2$	=	54
$\mathrm{c}_2^2$	=	324
$\mathrm{c}_1\mathrm{c}_3$	=	-90
$\mathrm{c}_4$	=	369

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		49		267		49		0
	0		0		0		0
		0		1		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = 0	0	0
$\mathrm{HH}^{ 2 }$ = 116	0	116	0
$\mathrm{HH}^{ 3 }$ = 0	0	0	0	0
$\mathrm{HH}^{ 4 }$ = 238	0	0	231	0	7
$\mathrm{HH}^{ 5 }$ = 0		0	0	0	0
$\mathrm{HH}^{ 6 }$ = 14			0	14	0
$\mathrm{HH}^{ 7 }$ = 0				0	0
$\mathrm{HH}^{ 8 }$ = 0					0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$

$\displaystyle\frac{ 1 + 2t + 3t^{2} + 2t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(747)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(736)

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 693t - 1 \right) \left( 36t + 1 \right) ^{4}D^{4} + 18 t \left( 13860t + 61 \right) \left( 36t + 1 \right) ^{3}D^{3} + 9 t \left( 36t + 1 \right) ^{2} \left( 3492720t^{2} + 57672t + 77 \right)D^{2} + 144 t \left( 36t + 1 \right) \left( 11226600t^{3} + 377622t^{2} + 2754t + 1 \right)D + 15552 t ^{2} \left( 1796256t^{3} + 98496t^{2} + 1605t + 7 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁶P6

Show multiplicities

Sector dimension: 5
Spectral radius: $693$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$693$	1	1
$-36$	4	4

Characteristic polynomial

$$\begin{equation}\left(\lambda + 36\right)^{4} \left(\lambda - 693\right)\notag\end{equation}$$

Minimal polynomial